We study the epidemic spreading on spatial networks where the probability that two nodes are connected decays with their distance as a power law. As the exponent of the distance dependence grows, model networks smoothly transition from the random network limit to the regular lattice limit. We show that despite keeping the average number of contacts constant, the increasing exponent hampers the epidemic spreading by making long-distance connections less frequent. The spreading dynamics is influenced by the distance-dependence exponent as well and changes from exponential growth to power-law growth. The observed power-law growth is compatible with recent analyses of empirical data on the spreading of COVID-19 in numerous countries.
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